New geodesics found that are not tight but still have useful properties.
problem Understanding geodesics in curve complexes and Teichmüller spaces.
method Introducing and studying weak tight geodesics with canonical constructions.
result Found examples of weak tight geodesics with gaps between them.
Geodesic mappings found in a specific type of warped product manifolds.
problem Characterizing geodesic mappings in a particular class of Roter spaces.
method Identifying a specific class of Roter type warped product manifolds and proving geodesic mappings onto another Roter type warped product manifold.
result Both geodesically related manifolds are pseudosymmetric of constant type.
Geodesics in mapping class group show unexpected behaviors.
problem Understanding geodesics and quasi-geodesics in mapping class groups.
method Constructing explicit examples and analyzing projections.
result Geodesics in mapping class group can have unexpected properties, e.g., not being quasi-geodesics.
In this paper we study fundamental properties of geodesic mappings with respect to the smoothness class of metrics. We show that geodesic mappings preserve the smoothness class of metrics. We study geodesic mappings of Einstein spaces.
New metric spaces for geodesic rays in cohomology classes.
problem Constructing geodesic rays in cohomology classes with finite energy.
method Introduced a chordal metric and proved geodesic properties.
result Found a characterization of geodesic rays in terms of test curves.
We parametrize the commensurability classes of curves on Shimura surfaces that are totally geodesic, i.e., the commensurability classes of so-called C-Fuchsian subgroups. In particular, if a Shimura surface contains one commensurability class of totally geodesic curves, it contains infinitely many.
We present a new equation with respect to a unit vector field on Riemannian manifold Mn such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasaki metric and apply it to some classes of unit vector fields. We introduce a class of covariantly normal unit vector fields and pro…
Mapping class group action on geodesic rays in hyperbolic surfaces is wandering.
problem Understanding the dynamics of mapping class group actions on geodesic rays.
method Analyzing the action of the mapping class group on the space of geodesic rays.
result The action is almost everywhere wandering.
Classifies and describes hypersurfaces in Siklos spacetimes.
problem Characterizing hypersurfaces in Siklos spacetimes.
method Classification and description of totally geodesic and parallel hypersurfaces.
result A large class of minimal hypersurfaces is described.
We present a complete description of a class of linearizable planar geodesic webs which contain a parallelizable 3-subweb.
We show that the strong asymptotic class of Weil-Petersson (WP) geodesics with narrow end invariant and bounded annular coefficients is determined by the forward ending lamination. This generalizes the Recurrent Ending Lamination Theorem of Brock-Masur-Minsky. As an application we provide a symbolic condition for diver…
Study Einstein Lie groups and geodesic orbit manifolds, finding some are not geodesic orbit.
problem Characterizing Einstein Lie groups and geodesic orbit manifolds.
method Characterizing GimesK-invariant geodesic orbit metrics on Lie groups G for regular subgroups K. result Extensive classes of compact simple Einstein Lie groups are not geodesic orbit manifolds.
Study confirms geodesic connectivity and rooftop envelopes in complex Monge-Ampère equation domains.
problem Confirming geodesic connectivity and rooftop envelopes in complex Monge-Ampère equation domains.
method Examined geodesics and plurisubharmonic envelopes within the Cegrell classes on bounded hyperconvex domains.
result Affirmative answer to a longstanding open question about geodesic connectivity and rooftop envelopes.
Polynomial decay of correlations shown for curved surfaces.
problem Analyzing geodesic flows on curved surfaces.
method Proving polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
result Polynomial decay of correlations for geodesic flows on nonpositively curved surfaces.
Polynomial-time algorithm for curve graph geodesics.
problem Computing distances and properties of mapping classes.
method Polynomial-time algorithm for geodesics in curve graphs.
result Asymptotic translation length, Nielsen-Thurston type, and canonical curve system can be computed in polynomial time.
Random walks on mapping class group lead to recurrent geodesics in quadratic differentials.
problem Understanding recurrence of geodesics in quadratic differentials.
method Analyzing random walks on mapping class group with specific properties.
result Recurrence of geodesics in the thick part of the principal stratum of quadratic differentials.
Random walks on mapping class groups identified with geodesic laminations.
problem Understanding random walks on mapping class groups.
method Electrification of curve graph, identifying Poisson boundary, using geodesic laminations.
result Random walk on mapping class group identified with geodesic laminations.
Counting hyperbolic multi-geodesics with individual component lengths.
problem Counting hyperbolic multi-geodesics with specific component lengths.
method Unified geometric and topological techniques, combining Mirzakhani's results and Margulis's ideas.
result Asymptotic polynomial counts of multi-geodesics in mapping class group orbits, generalizing Wolpert's conjecture.
The paper proves that certain geodesics pass through timelike poles on specific Lorentzian 2-tori.
problem Existence of closed timelike geodesics through timelike poles on class A Lorentzian 2-tori.
method Study of isometries on globally hyperbolic planes to prove the existence of geodesics.
result Existence of closed timelike geodesics through timelike poles in the interior of the stable time cone.
New distance function proves rigidity in geodesic lamination space.
problem Proving rigidity in geodesic lamination space.
method Introduced left Hausdorff distance function and proved rigidity result.
result Extended mapping class group is isomorphic to bijections preserving left Hausdorff convergence.
Random walks on mapping classes lead to pseudo-Anosov maps in the principal stratum.
problem Understanding the behavior of random walks on mapping class groups.
method Analyzing random walks with specific subgroup properties and pseudo-Anosov maps.
result Almost every infinite sample path of random walks contains pseudo-Anosov maps with invariant geodesics in the principal stratum.
Study geodesic string counts on Riemann-Finsler manifolds, linking to KAM theory.
problem Counting geodesic strings on Riemann-Finsler manifolds.
method Using Fuller index and KAM theory, derive product formula for counts.
result Arithmetic constraints on geodesic string counts and existence of negative curvature metrics.
Study shows mean curvature flows converge to a geodesic graph over a totally geodesic hypersurface.
problem Mean curvature flows in warped product manifolds with closed hypersurfaces.
method Investigation of mean curvature flows in specific warped product manifolds with conditions on warping function and Ricci curvature.
result Existence and convergence of mean curvature flows for certain initial hypersurfaces.
We show that both Teichmuller space (with the Teichmuller metric) and the mapping class group (with a word metric) have geodesic divergence that is intermediate between the linear rate of flat spaces and the exponential rate of hyperbolic spaces. For every two geodesic rays in Teichmuller space, we find that their dive…
The study explores different definitions of geodesics in sub-Riemannian geometry.
problem Understanding geodesics in sub-Riemannian geometry and their equivalence.
method Review of three variational definitions of geodesics and three definitions of straightest curves.
result Shortest geodesics coincide with straightest geodesics in some sub-Riemannian manifolds.
The paper defines a complete geodesic metric for high energy spaces in Kähler manifolds.
problem Defining a metric for high energy spaces in Kähler manifolds.
method Endowing the high energy space with a metric that makes it a complete geodesic metric space.
result The geodesic metric space (Ep(X,θ),dp) is uniformly convex for p>1. New boundary for geodesic spaces captures Poisson boundary of mapping class groups.
problem Capturing the Poisson boundary of mapping class groups.
method Constructing a quasi-isometric invariant boundary for proper geodesic spaces.
result The Poisson boundary of mapping class groups can be realized on the κ-Morse boundary.
Study geodesic discs with boundary length bounds, finding their closure in metric space.
problem Geodesic discs with boundary length constraints in metric spaces.
method Investigate closure in Gromov-Hausdorff space, relate to disc retracts.
result Closure of geodesic discs is related to disc retracts in metric spaces.
Geodesic connectedness proved for statistical manifolds with divisible cubic forms.
problem Geodesic connectedness of affine connections on statistical manifolds with divisible cubic forms.
method Analogy with Hopf-Rinow theorem in Riemannian geometry, establishing geodesic completeness.
result Geodesic connectedness established for statistical manifolds with divisible cubic forms.
Even-dimensional orbifolds with all closed geodesics cover manifolds.
problem Understanding geodesics in orbifolds with singularities.
method Characterization of orientable manifolds among orientable orbifolds using characteristic classes.
result Odd-dimensional orbifolds with all geodesics closed are covered by manifolds.
Study geodesic distances and convexity in contact sets.
problem Understanding geodesic distances and convexity in contact sets.
method Extending results on quasi-psh functions and big cohomology classes, studying Monge-Ampère measures on contact sets.
result Convexity of the K-energy in big and nef cohomology classes.
Geodesics found in a metric space of m-subharmonic functions.
problem Metric structure on energy class of m-subharmonic functions.
method Inspired by Kähler geometry, introduced a metric structure and studied metric convergence.
result Geodesics constructed in a subspace of the complete metric space.
Study shows continuity of renormalized volume for geometrically convergent hyperbolic structures.
problem Continuity of renormalized volume under geometric limits.
method Extended renormalized volume concept for geometrically finite hyperbolic 3-manifolds and showed continuity for geometrically convergent sequences.
result Renormalized volume attains its minimum at the geodesic class.
Proves C1,1 regularity for complex Monge-Ampère equations and geodesic rays.
problem Complex Monge-Ampère equations on compact Kähler manifolds with degenerate cohomology.
method Proves C1,1 estimate for solutions. result Local C1,1 regularity of geodesic rays and quasi-psh envelopes. The local-to-global property is proven for Morse quasi-geodesics in various groups.
problem Proving local-to-global properties for Morse quasi-geodesics in different groups.
method Developing a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces.
result Generalization of combination theorems for stable subgroups of various groups.
Geodesic flow on submanifolds is shown to be Ck−1.
problem Regularity of geodesic flow on submanifolds.
method Analysis of Ck submanifolds with k≥2. result Geodesic flow and exponential map are Ck−1. New approach constructs symplectic structure on pseudo-Riemannian geodesics.
problem Dimensional mismatch in classical symplectic structure construction for pseudo-Riemannian geodesics.
method Directly constructs geodesic space as quotient, introduces conformal co-symplectic structure.
result Conformal co-symplectic structure globally describes geometric distribution of geodesics.
The paper counts orbits of curves with up to K self-intersections on a surface.
problem Counting orbits of curves with self-intersections on a surface.
method Combinatorial approach to studying geodesics on surfaces.
result Developed a new combinatorial method to study geodesics on surfaces.
Researchers prove a property for a specific group class, leading to Wasserstein geodesic continuity.
problem Proving a measure contraction property for generalized H-type Carnot groups.
method Analyzing H-type Carnot groups of rank k and dimension n to establish the MCP(K,N) condition. result Generalized H-type Carnot groups satisfy MCP(K,N) with K≤0 and N≥k+3(n−k), matching geodesic dimension. Study geodesics entering a fixed cusp neighborhood multiple times.
problem Understanding geodesics entering a specific cusp neighborhood multiple times.
method Investigate reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
result Characterized the class of reciprocal geodesics entering a fixed cusp neighborhood a fixed number of times.
Abstract Coxeter groups have growth rates that are Perron numbers.
problem Understanding growth rates of Coxeter groups.
method Defined a class of Coxeter groups, ∞--spanned, and analyzed their growth rates. result For ∞--spanned Coxeter groups, geodesic growth rate strictly dominates word growth rate and appears to be a Perron number. In this paper we analyze and classify the totally geodesic subspaces of finite volume quaternionic hyperbolic orbifolds and their generalizations, locally symmetric orbifolds arising from irreducible lattices in Lie groups of the form $(\mathbf{Sp}_{2n}(\mathbb{R}))^q \times \prod_{i=1}^r \mathbf{Sp}(p_i,n-p_i) \times …
New Teichmüller geodesic rays found with unique foliations.
problem Capturing generic directions in Teichmüller space.
method Using Chaika-Masur-Wolf and Durham-Zalloum work.
result First sublinearly-Morse geodesic rays with minimal non-uniquely ergodic foliations.
Study geodesic orbit metrics on specific homogeneous spaces.
problem Characterize geodesic orbit spaces in a class of homogeneous bundles.
method Analyze geodesic orbit spaces of compact Lie groups with semisimple subgroups.
result Identify conditions for a metric to be geodesic orbit.
Study on geodesics and dihedral groups in lattices.
problem Growth and distribution of conjugacy classes of dihedral subgroups.
method Generalizing earlier work on reciprocal geodesics, proving equidistribution.
result Reciprocal geodesics are equidistributed in the unit tangent bundle.
Estimates geodesics on surfaces without conjugate points.
problem Counting geodesics on surfaces without conjugate points.
method Margulis-type asymptotic estimates.
result Asymptotic estimates for geodesics on surfaces.
Classifies special hypersurfaces in Gödel spacetimes.
problem Characterizing hypersurfaces in Gödel spacetimes.
method Classification of parallel and totally geodesic hypersurfaces.
result Identified specific types of hypersurfaces in Gödel spacetimes.
A projection maps geodesic currents to Teichmüller space.
problem Mapping geodesic currents to Teichmüller space.
method Equivariant, length-minimizing projection from filling currents to Teichmüller space.
result The projection is well-behaved and maps geodesic currents to Teichmüller space.